Spiking model to learn arbitrary multiple transformations for a self-realizing network

ABSTRACT

A neural network, wherein a portion of the neural network comprises: a first array having a first number of neurons, wherein the dendrite of each neuron of the first array is provided for receiving an input signal indicating that a measured parameter gets closer to a predetermined value assigned to said neuron; and a second array having a second number of neurons, wherein the second number is smaller than the first number, the dendrite of each neuron of the second array forming an excitatory STDP synapse with the axon of a plurality of neurons of the first array; the dendrite of each neuron of the second array forming an excitatory STDP synapse with the axon of neighboring neurons of the second array.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a non-provisional and claims priority of U.S.provisional application No. 61/799,883, filed Mar. 15, 2013, which isincorporated herein as though set forth in full.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with support from the United States Governmentunder contract number HR0011-09-C-0001 (SyNAPSE) awarded by the DefenseAdvanced Research Project Agency (DARPA). The United States Governmenthas certain rights in the invention.

BACKGROUND OF THE DISCLOSURE

1. Field of the Disclosure

The present disclosure relates to neural networks; for examplecomputer-implemented, as well as to methods for programming suchnetworks. In particular, the present disclosure relates to a faulttolerant neural network capable of learning arbitrary multipletransformations, and a method of programming said neural network.

2. Background

Sensory perception and action are interdependent. In humans and otherspecies, a behavior may be triggered by an ongoing situation andreflects a being's immediate environmental conditions. This type ofbehavior is often referred to as stimulus-response reflexes. Theinterdependency between stimulus and response creates an actionperception cycle in which a novel stimulus triggers actions that lead toa better perception of itself or its immediate environmental conditionand the cycle continues.

Human behavior is much more flexible than exclusive control bystimulus-response cycles. One attribute of intelligent-based systems isthe ability to learn new relations between environmental conditions andappropriate behavior during action perception cycles. The primary modeof communication between neurons in the brain is encoded in the form ofimpulses, action potentials or spikes. The brain is composed of billionsof neural cells, which are noisy, imprecise and unreliable analogdevices. The neurons are complex adaptive structures that makeconnections between each other via synapses. A synapse has a presynapticportion, comprising the axon of a neuron, inputing a spike into thesynapse, and a postsynaptic portion comprising the dendrite of a neuron,sensitive to the spike being received in the synapse. The synapses maychange their function dramatically depending upon the spiking activityof the neurons on either side of the synapse. The synapse includes anadaptation mechanism that adjusts the weight, or gain, of the synapseaccording to a spike timing dependent plasticity (STDP) learning rule.

Under the STDP rule, if an input spike to a neuron tends, on average, tooccur immediately before that neuron's output spike, then thatparticular input is made somewhat stronger. On another hand, if an inputspike tends, on average, to occur immediately after an output spike,then that particular input is made somewhat weaker hence:“spike-timing-dependent plasticity”. Thus, inputs that might be thecause of the post-synaptic neuron's excitation are made even more likelyto contribute in the future, whereas inputs that are not the cause ofthe post-synaptic spike are made less likely to contribute in thefuture. The process continues until a subset of the initial set ofconnections remains, while the influence of all others is reduced to 0.Since a neuron produces an output spike when many of its inputs occurwithin a brief period the subset of inputs that remain are those thattended to be correlated in time. In addition, since the inputs thatoccur before the output are strengthened, the inputs that provide theearliest indication of correlation eventually become the final input tothe neuron.

Brain architectures composed of assemblies of interacting neurons andsynapses with STDP can solve complex tasks and exhibit complex behaviorsin real-time and with high precision but with very low power. However,modeling such activity in a physical network is complex.

Neural networks using analog and digital circuitry andcomputer-implemented methods have been discussed to implement a STDPlearning rule. However, current models do not have the capacity to betolerant to faults (i.e., to partial absence of sensory or motor inputsignals) introduced either from the beginning of the learning process orafter some initial learning has taken place. Accordingly, the knownsystems that implement a STDP learning rule are incapable of learningfor example arbitrary multiple transformations in a fault tolerantfashion.

Several examples of communication systems that have experienced theabove described communication issues include T. P. Vogels, K. Rajan andL. F. Abbott, “Neural Network Dynamics,” Annual Review Neuroscience,vol. 28, pp. 357-376, 2005; W. Gerstner and W. Kistler, Spiking NeuronModels—Single Neurons, Populations, Plasticity, Cambridge UniversityPress, 2002; H. Markram, J. Lubke, M. Frotscher, & B. Sakmann,“Regulation of synaptic efficacy by coincidence of postsynaptic APs andEPSPs,” Science, vol. 275, pp. 213-215, 1997; Bi, G. Q., & M. Poo,“Activity-induced synaptic modifications in hippocampal culture:dependence on spike timing, synaptic strength and cell type,” J.Neuroscience. vol. 18, pp. 10464-10472, 1998; J. C. Magee and D.Johnston, “A synaptically controlled, associative signal for Hebbianplasticity in hippocampal neurons,” Science vol. 275, pp. 209-213, 1997;S. Song, K. D. Miller and L. F. Abbott, “Competitive Hebbian LearningThrough Spike-Timing Dependent Synaptic Plasticity,” NatureNeuroscience, vol. 3 pp. 919-926, 2000; A. P. Davison and Y. Fregnac,“Learning Cross-Modal Spatial Transformations through Spike-TimingDependent Plasticity,” Journal of Neuroscience, vol. 26, no. 2, pp.5604-5615, 2006; Q. X. Wu, T. M. McGinnity, L. P. Maguire, A.Belatreche, B. Glackin, “2D co-ordinate transformation based on aspike-timing dependent plasticity learning mechanism,” Neural Networks,vol. 21, pp. 1318-1327, 2008; Q. X. Wu, T. M. McGinnity, L. P. Maguire,A. Belatreche, B. Glackin; “Processing visual stimuli using hierarchicalspiking neural networks,” International Journal of Neurocomputing, vol.71, no. 10, pp. 2055-2068, 2008. Each of the above references is herebyincorporated by reference in its entirety.

FIG. 1 illustrates a network model described in the above referenceentitled “Learning Cross-Modal Spatial Transformations throughSpike-Timing Dependent Plasticity”. FIG. 1 shows a neural network thatreceives in input the angle θ at the joint of an arm with 1 Degree ofFreedom (df) and the position x of the end of the arm, in avision-centered frame of reference. After a learning phase the neuralnetwork becomes capable of outputting x based on the angle θ at thejoint. The neural network 10 comprises a first one-dimension array 12 ofinput neurons 14 that each generate spikes having a firing rate thatincreases as a function of the angle θ getting closer to an angleassigned to the neuron. FIG. 1 illustrates the firing rate FR of all theneurons 14 of array 12 for a given value of the angle θ. The neuralnetwork 10 further comprises a second one-dimension array 16 of inputneurons 18 that each generate spikes having a firing rate that increasesas a function of the position x getting closer to a predetermined valueassigned to the neuron. FIG. 1 illustrates the firing rate FR of all theneurons 18 of array 16 for a given value of the position x. The neuralnetwork 10 comprises a third one-dimension array 20 of neurons 22.

Connections are initially all-to-all (full connection) from the neurons14 to the neurons 22, and the strength of the connections is subject tomodification by STDP. Connections from the neurons 18 to the neurons 22are one to one. The strength of these non-STPD (or non-plastic)connections is fixed.

After a learning phase where stimuli corresponding to random angle θ andtheir equivalent position x are sent to array 20, array 16 ceases toprovide input to the array 20, and array 20 outputs a position x inresponse to a based on the angle θ at the joint. FIG. 1 illustrates thefiring rate FR output by the neurons 22 of array 20 in response to agiven value of the angle θ after the learning phase.

FIG. 2 illustrates in schematic form the neural network 10 of FIG. 1 andshows input array/layer 12 fully connected to output array/layer 20 andtraining array/layer 16 connected on-to-one to output array/layer 20.

FIG. 3 schematizes a neural network 30 such as disclosed in the Wu etal. reference above. The neural network 30 comprises a training layer 16connected one-to-one to an output layer 20 as detailed in FIG. 1.Further, neural network 30 comprises two input layers 12 topographicallyconnected in input of a network layer 32, the network layer 32 beingfully connected in input of output layer 20. As outlined above, theneural networks of FIGS. 1-3 are not tolerant to faults such as apartial absence of sensory or motor input signals, introduced eitherfrom the beginning of the learning process or after some initiallearning has taken place.

There exists a need for neural networks that would be tolerant to fault.

SUMMARY

An embodiment of the present disclosure comprises a neural network,wherein a portion of the neural network comprises: a first array havinga first number of neurons, wherein the dendrite of each neuron of thefirst array is provided for receiving an input signal indicating that ameasured parameter gets closer to a predetermined value assigned to saidneuron; a second array having a second number of neurons, the dendriteof each neuron of the second array forming an excitatory STDP synapsewith the axon of a plurality of neurons of the first array; the dendriteof each neuron of the second array forming an excitatory STDP synapsewith the axon of neighboring neurons of the second array.

According to an embodiment of the present disclosure, the second numberis smaller than the first number.

According to an embodiment of the present disclosure, the second arrayfurther comprises a third number of interneurons distributed among theneurons of the second array, wherein the third number is smaller thanthe second number, wherein: the axon of each neuron of the second arrayforms an excitatory STDP synapse with the dendrite of the neighboringinterneurons of the second array; and the axon of each interneuron ofthe second array forms an inhibitory STDP synapse with the dendrite ofthe neighboring neurons and interneurons of the second array.

According to an embodiment of the present disclosure, the dendrite ofeach neuron of the first array is provided for receiving an input signalhaving a rate that increases when a measured parameter gets closer to apredetermined value assigned to said neuron.

An embodiment of the present disclosure comprises a neural networkhaving a first and a second neural network portions as described above,as well as a third array having a fourth number of neurons and a fifthnumber of interneurons distributed among the neurons of the third array,wherein the fifth number is smaller than the fourth number, wherein: theaxon of each neuron of the third array forms an excitatory STDP synapsewith the dendrite of the neighboring interneurons of the third array;and the axon of each interneuron of the third array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the third array; wherein the axon of each neuron of thesecond array of the first neural network portion forms an excitatorySTDP synapse with the dendrite of a plurality of neurons of the thirdarray; and wherein the axon of each neuron of the second array of thesecond neural network portion forms an excitatory STDP synapse with thedendrite of a plurality of neurons of the third array.

According to an embodiment of the present disclosure, the third arraycomprises rows and columns of neurons, wherein the axon of each neuronof the second array of the first neural network portion forms anexcitatory STDP synapse with the dendrite of a plurality of neurons of arow of the third array; and wherein the axon of each neuron of thesecond array of the second neural network portion forms an excitatorySTDP synapse with the dendrite of a plurality of neurons of a column ofthe third array.

According to an embodiment of the present disclosure, the neural networkcomprises a third neural network portion as described above, as well asa fourth array having a second number of neurons and a third number ofinterneurons distributed among the neurons of the fourth array, wherein:the axon of each neuron of the fourth array forms an excitatory STDPsynapse with the dendrite of the neighboring interneurons of the fourtharray; and the axon of each interneuron of the fourth array forms aninhibitory STDP synapse with the dendrite of the neighboring neurons andinterneurons of the fourth array; wherein the dendrite of each neuron ofthe fourth array forms an excitatory STDP synapse with the axon of aplurality of neurons of the third array; and wherein the dendrite ofeach neuron of the fourth array forms an excitatory non-STDP synapsewith the axon of a corresponding neuron of the second array of the thirdneural network.

According to an embodiment of the present disclosure, the input signalsto the first and second neural network portions relate to variableparameters that are to be correlated to the input signals to the thirdneural network.

According to an embodiment of the present disclosure, the first array ofneurons comprises first and second sub-arrays of neurons provided forreceiving input signals related to first and second measured parameters,respectively.

According to an embodiment of the present disclosure, the second arraycomprises rows and columns of neurons; wherein the axon of each neuronof the first sub-array of neurons forms an excitatory STDP synapse withthe dendrite of a plurality of neurons of a row of the second array; andwherein the axon of each neuron of the second sub-array of neurons formsan excitatory STDP synapse with the dendrite of a plurality of neuronsof a column of the second array.

According to an embodiment of the present disclosure, the second arrayfurther comprises a third number of interneurons distributed among theneurons of the second array, wherein the third number is smaller thanthe second number, wherein the axon of each neuron of the second arrayforms an excitatory STDP synapse with the dendrite of the neighboringinterneurons of the second array; and the axon of each interneuron ofthe second array forms an inhibitory STDP synapse with the dendrite ofthe neighboring neurons and interneurons of the second array.

According to an embodiment of the present disclosure, the neural networkfurther comprises: a third array having a fourth number of neurons and afifth number of interneurons distributed among the neurons of the thirdarray, wherein the fifth number is smaller than the fourth number,wherein: the axon of each neuron of the third array forms an excitatorySTDP synapse with the dendrite of the neighboring interneurons of thethird array; and the axon of each interneuron of the third array formsan inhibitory STDP synapse with the dendrite of the neighboring neuronsand interneurons of the third array; wherein the dendrite of each neuronof the third array forms an excitatory STDP synapse with the axon ofeach neuron of the second array.

According to an embodiment of the present disclosure, the neural networkcomprises as many neurons as the third array of neurons, wherein thedendrite of each neuron of the fourth array is provided for receiving aninput signal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron; wherein the axon of eachneuron of the fourth array forms an excitatory non-STDP synapse with thedendrite of a corresponding neuron of the third array.

According to an embodiment of the present disclosure, the input signalsto the first and second sub-arrays of neurons relate to variableparameters that are to be correlated to the input signals to the fourtharray.

According to an embodiment of the present disclosure, the fourth arrayof neurons is a sub-array of neurons of a further neural network asdescribed above.

Another embodiment of the present disclosure comprises a method ofprogramming a neural network, the method comprising: providing a firstneural network portion comprising a first array having a first number ofneurons and a second array having a second number of neurons, whereinthe second number is smaller than the first number, the dendrite of eachneuron of the second array forming an excitatory STDP synapse with theaxon of a plurality of neurons of the first array; the dendrite of eachneuron of the second array forming an excitatory STDP synapse with theaxon of neighboring neurons of the second array; and providing to thedendrite of each neuron of the first array an input signal indicatingthat a measured parameter gets closer to a predetermined value assignedto said neuron.

According to an embodiment of the present disclosure, the method furthercomprises providing the second array with a third number of interneuronsdistributed among the neurons of the second array, wherein the thirdnumber is smaller than the second number, wherein: the axon of eachneuron of the second array forms an excitatory STDP synapse with thedendrite of the neighboring interneurons of the second array; and theaxon of each interneuron of the second array forms an inhibitory STDPsynapse with the dendrite of the neighboring neurons and interneurons ofthe second array.

According to an embodiment of the present disclosure, the methodcomprises providing the dendrite of each neuron of the first array withan input signal having a rate that increases when a measured parametergets closer to a predetermined value assigned to said neuron.

According to an embodiment of the present disclosure, the methodcomprises: providing a second neural network portion having the samestructure as the first neural network portion; and providing a thirdarray having a fourth number of neurons and a fifth number ofinterneurons distributed among the neurons of the third array, whereinthe fifth number is smaller than the fourth number, wherein: the axon ofeach neuron of the third array forms an excitatory STDP synapse with thedendrite of the neighboring interneurons of the third array; and theaxon of each interneuron of the third array forms an inhibitory STDPsynapse with the dendrite of the neighboring neurons and interneurons ofthe third array; wherein the axon of each neuron of the second array ofthe first neural network portion forms an excitatory STDP synapse withthe dendrite of a plurality of neurons of the third array; and whereinthe axon of each neuron of the second array of the second neural networkportion forms an excitatory SILT synapse with the dendrite of aplurality of neurons of the third array; and providing to the dendriteof each neuron of the first array of the second neural network portionan input signal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron.

According to an embodiment of the present disclosure, the methodcomprises: providing a third neural network portion having the samestructure as the first neural network portion; providing a fourth arrayhaving a second number of neurons and a third number of interneuronsdistributed among the neurons of the fourth array, wherein: the axon ofeach neuron of the fourth array forms an excitatory STDP synapse withthe dendrite of the neighboring interneurons of the fourth array; andthe axon of each interneuron of the fourth array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the fourth array; wherein the dendrite of each neuron ofthe fourth array forms an excitatory STDP synapse with the axon of aplurality of neurons of the third array; and wherein the dendrite ofeach neuron of the fourth array forms an excitatory non-STDP synapsewith the axon of a corresponding neuron of the second array of the thirdneural network; and providing to the dendrite of each neuron of thefirst array of the third neural network portion an input signalindicating that a measured parameter gets closer to a predeterminedvalue assigned to said neuron.

According to an embodiment of the present disclosure, the input signalsto the first and second neural network portions relate to variableparameters that are to be correlated to the input signals to the thirdneural network portion.

According to an embodiment of the present disclosure, said providing tothe dendrite of each neuron of the first array an input signalindicating that a measured parameter gets closer to a predeterminedvalue assigned to said neuron comprises: providing to the dendrite ofeach neuron of a first subset of neurons of the first array an inputsignal indicating that a first measured parameter gets closer to apredetermined value assigned to said neuron; providing to the dendriteof each neuron of a second subset of neurons of the first array an inputsignal indicating that a second measured parameter gets closer to apredetermined value assigned to said neuron.

According to an embodiment of the present disclosure, said providing asecond array having a second number of neurons comprises providing asecond array having rows and columns of neurons, wherein the axon ofeach neuron of the first subset of neurons of the first array forms anexcitatory STDP synapse with the dendrite of a plurality of neurons of arow of the second array; and wherein the axon of each neuron of thesecond subset of neurons of the first array forms an excitatory STDPsynapse with the dendrite of a plurality of neurons of a column of thesecond array.

According to an embodiment of the present disclosure, the method furthercomprises providing the second array with a third number of interneuronsdistributed among the neurons of the second array, wherein the thirdnumber is smaller than the second number, wherein the axon of eachneuron of the second array forms an excitatory STDP synapse with thedendrite of the neighboring interneurons of the second array; and theaxon of each interneuron of the second array forms an inhibitory STDPsynapse with the dendrite of the neighboring neurons and interneurons ofthe second array.

According to an embodiment of the present disclosure, the methodcomprises: providing a third array having a fourth number of neurons anda fifth number of interneurons distributed among the neurons of thethird array, wherein the fifth number is smaller than the fourth number,wherein the axon of each neuron of the third array forms an excitatorySTDP synapse with the dendrite of the neighboring interneurons of thethird array; and the axon of each interneuron of the third array formsan inhibitory STDP synapse with the dendrite of the neighboring neuronsand interneurons of the third array; wherein the dendrite of each neuronof the third array forms an excitatory STDP synapse with the axon ofeach neuron of the second array; and providing a fourth array comprisingas many neurons as the third array of neurons, wherein the dendrite ofeach neuron of the fourth array is provided for receiving an inputsignal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron; and wherein the axon ofeach neuron of the fourth array forms an excitatory non-STDP synapsewith the dendrite of a corresponding neuron of the third array; themethod further comprising providing to the dendrite of each neuron ofthe fourth array an input signal indicating that a measured parametergets closer to a predetermined value assigned to said neuron; whereinthe input signals to the first and second subset of neurons relate tovariable parameters that are to be correlated to the input signals tothe fourth array.

Another embodiment of the present disclosure comprises a method ofdecoding an output of a neural network having first and second neuralnetwork portions as detailed above; the method comprising: providing thefirst arrays of the first and second neural network portions with firstand second input signals having a rate that increases when a measuredparameter gets closer to a predetermined value assigned to the neuronsof said first arrays; assigning to each neuron of the fourth array ofneurons an incremental position value comprised between 1 and N, N beingthe number of neurons of the fourth array; at any given time, measuringthe firing rate of each neuron of the fourth array; and estimating theoutput of the neural network, at said any given time, as correspondingto the neuron of the fourth array having a position value equal to adivision of the sum of the position value of each neuron of the fourtharray, weighted by its firing rate at said any given time, by the sum ofthe firing rates of each neuron of the fourth array at said any giventime.

According to an embodiment of the present disclosure, the methodcomprises, if the neurons of the middle of the fourth array have nullfiring rates, assigning to the neurons of lower position value aposition value increased by the value N.

Another embodiment of the present disclosure comprises a method ofdecoding an output of a neural network having first and secondsub-arrays of neurons as disclosed above; the method comprising:providing the first and second sub-arrays of neurons with first andsecond input signals having a rate that increases when a measuredparameter gets closer to a predetermined value assigned to the neuronsof said first and second sub-arrays of neurons; assigning to each neuronof the third array of neurons an incremental position value comprisedbetween 1 and N, N being the number of neurons of the third array; atany given time, measuring the firing rate of each neuron of the thirdarray; and estimating the output of the neural network, at said anygiven time, as corresponding to the neuron of the third array having aposition value equal to a division of the sum of the position value ofeach neuron of the third array, weighted by its firing rate at said anygiven time, by the sum of the firing rates of each neuron of the thirdarray at said any given time.

According to an embodiment of the present disclosure, the methodcomprises, if the neurons of the middle of the third array have nullfiring rates, assigning to the neurons of lower position value aposition value increased by the value N.

An embodiment of the present disclosure comprises a neural network thatincludes a plurality of input channels; an intermediate layer of neuronsincluding a plurality of recurrent connections between a plurality ofthe neurons; a plurality of inhibitor interneurons connected to theintermediate layer of neurons; a plurality of first connectionsconfigured to connect the intermediate layer of neurons to a predictionlayer; and a plurality of second connections configured to connect theprediction layer to an output layer.

According to an embodiment of the present disclosure, the output layeris configured to be connected to a further layer of neurons, and thefurther layer of neurons may be connected to one or more additionalprediction layers by one or more connections. The one or more additionalprediction layers may be configured to be connected to one or moreadditional circuits. The intermediate layer of neurons may be connectedto the plurality of inhibitor interneurons by a plurality of electricalsynapses. The input channels may provide a spike train to the firstlayer of neurons.

An embodiment of the present disclosure comprises a non-transitorycomputer-useable storage medium for signal delivery in a systemincluding multiple circuits, said medium having a computer-readableprogram, wherein the program upon being processed on a computer causesthe computer to implement the steps of: receiving at a first layer ofneurons a spike train; transferring a plurality of inhibitorinterneurons to the first layer of neurons; passing the first layer ofneurons, by a plurality of first connections, to a prediction layer; andcoupling the prediction layer to an output circuit by a plurality ofsecond connections.

An embodiment of the present disclosure comprises a method of signaldelivery in a system including a plurality of input channels includingreceiving at a first layer of neurons a spike train; transferring aplurality of inhibitor interneurons to the first layer of neurons;passing the first layer of neurons, by a plurality of first connections,to a prediction layer; and coupling the prediction layer to an outputcircuit by a plurality of second connections.

BRIEF DESCRIPTION OF THE FIGURES

The disclosure may be better understood by referring to the followingfigures. The components in the figures are not necessarily to scale,emphasis instead being placed upon illustrating the principles of thedisclosure. In the figures, like reference numerals designatecorresponding parts throughout the different views.

FIG. 1 illustrates a known neural network model.

FIG. 2 is a schematic of model of FIG. 1.

FIG. 3 is a schematic of another known neural network model.

FIG. 4 illustrates a portion of a neural network model according to anembodiment of the present disclosure.

FIG. 5 illustrates a portion of a neural network model according to anembodiment of the present disclosure.

FIG. 6 illustrates a portion of a neural network model according to anembodiment of the present disclosure.

FIG. 7 is a schematic of a neural network model according to anembodiment of the present disclosure.

FIG. 8 illustrates the application of a neural network model accordingto an embodiment of the present disclosure to a 2DL robotic arm.

FIG. 9 illustrates the synaptic conductances between various layersduring learning showing the emergence of topological organization ofconductances in the neural network model of FIG. 8.

FIGS. 10A-B illustrate the output of layer Le in response to inputs onlayers L₁ ¹ and L₁ ² of the neural network model of FIG. 8.

FIGS. 11A-C illustrate the incremental convergence of the neural networkmodel of FIG. 8 as a function of learning.

FIGS. 12A-D illustrate the incremental convergence of the neural networkmodel of FIG. 8 for Gaussian sparse connectivity and random sparseconnectivity.

FIGS. 13A-D illustrate the performances of the neural network model ofFIG. 8 for varying degrees of damaged neurons.

FIG. 14 is a schematic of a neural network model according to anembodiment of the present disclosure.

FIG. 15 is a schematic of a further embodiment of the neural networkmodel of FIG. 14.

FIG. 16 is a schematic of a further embodiment of the neural networkmodel of FIG. 14.

DETAILED DESCRIPTION

Each of the additional features and teachings disclosed below can beutilized separately or in conjunction with other features and teachingsto provide a computer-implemented device, system, and/or method for aneural network model to learn arbitrary multiple transformations for aself-realizing network. Representative examples of embodiments of thepresent disclosure, which examples utilize many of these additionalfeatures and teachings both separately and in combination, will now bedescribed in further detail with reference to the attached drawings. Thepresent detailed description is merely intended to teach a person ofskill in the art further details for practicing preferred aspects of thepresent teachings and is not intended to limit the scope of thedisclosure. Therefore, combinations of features and steps disclosed inthe following detail description may not be necessary to practiceembodiments of the present disclosure in the broadest sense, and areinstead taught merely to particularly describe representative examplesof the present teachings.

The following are expressly incorporated by reference in their entiretyherein: “Self-Organizing Spiking Neural Model for LearningFault-Tolerant Spatio-Motor Transformations,” IEEE Transactions onNeural Networks and Learning Systems, Vol. 23, No. 10, October 2012;U.S. patent application Ser. No. 13/679,727, filed Nov. 16, 2012, andentitled “Spike Domain Neuron Circuit with Programmable KineticDynamics, Homeostatic Plasticity and Axonal Delays;” U.S. patentapplication Ser. No. 13/415,812, filed on Mar. 8, 2012, andentitled“Spike Timing Dependent Plasticity Apparatus, System andMethod;” and U.S. patent application Ser. No. 13/708,823, filed on Dec.7, 2012, and entitled “Cortical Neuromorphic Network System and Method.”

Devices, methods, and systems are hereby described for a neural networkmodel; in particular a spiking model capable of learning arbitrarymultiple transformations for a self-realizing network (SRN). Thedescribed systems and methods may be used to develop self-organizingrobotic platforms (SORB) that autonomously discover and extract keypatterns during or from real world interactions. In some configurations,the interactions may occur without human intervention. The described SRNmay be configured for unmanned ground and air vehicles for intelligence,surveillance, and reconnaissance (ISR) applications.

FIG. 4 illustrates a portion of a neural network or neural network model40 according to an embodiment of the present disclosure. According to anembodiment of the present disclosure, an input array/layer 12 comprisesa first number of neurons 14. The dendrite of each neuron 14 of theinput array 12 is provided for receiving an input signal indicating thata measured parameter gets closer to a predetermined value assigned tosaid neuron. 11.

According to an embodiment of the present disclosure, the input signalsent to each neuron 14, relating to a measured parameter, has a ratethat increases when the measured parameter gets closer to apredetermined value assigned to said neuron. FIG. 4 shows the firingrate FR of the input signals at a given time, with respect to theposition value PV of the neurons 14. According to an embodiment of thepresent disclosure, the neurons are integrate and fire neurons, oroperate under a model of integrate and fire neurons and the neuralnetwork or neural network model is a spiking neural network or spikingneural network model.

According to an embodiment of the present disclosure, the portion ofneural network model 40 comprises an intermediate array/layer 42 havinga second number of neurons 44. According to an embodiment of the presentdisclosure, the second number is smaller than the first number.According to an embodiment of the present disclosure, the dendrite ofeach neuron 44 of the intermediate array forms an excitatory STDPsynapse with the axon of a plurality of neurons 14 of the input array12. According to an embodiment of the present disclosure, the dendriteof each neuron 44 of the intermediate array 42 can form STDP synapseswith the axon of 100 to 200 neurons 14 of the input array.

According to an embodiment of the present disclosure, the dendrite ofeach neuron 44 of the intermediate array 42 forms an excitatory STDPsynapse 46 with the axon of neighboring neurons 44 of the intermediatearray 42. According to an embodiment of the present disclosure,neighboring neurons can be a predetermined number of closest neurons inboth direction of the array. According to an embodiment of the presentdisclosure, the intermediate array 42 further comprises a third numberof interneurons 48 distributed among the neurons 44, wherein the thirdnumber is smaller than the second number. According to an embodiment ofthe present disclosure, the third number can be about one fourth of thesecond number. According to an embodiment of the present disclosure, theinterneurons 48 of an array are equally distributed among the neurons44, for example according to a periodic or pseudorandom scheme.According to an embodiment of the present disclosure, the axon of eachneuron 44 of the intermediate array 42 forms an excitatory STDP synapse50 with the dendrite of a neighboring interneuron 48 of the intermediatearray 42; and the axon of each interneuron 48 of the intermediate array42 forms an inhibitory STDP synapse 52 with the dendrite of neighboringneurons 44 and interneurons 48 of the intermediate array 42. Therecurrence in the intermediate layer enables a neural network or neuralnetwork model according to an embodiment of the present disclosure to befault-tolerant. This is because neurons in the intermediate layer thatdo not receive inputs from the input layer neurons may receive inputsfrom within the neurons in the intermediate layer. This allows thestructure to be able to interpolate the network activity despite theabsence of feedforward inputs.

FIG. 5 illustrates a portion of a neural network or neural network model60 according to an embodiment of the present disclosure. According to anembodiment of the present disclosure, the portion of neural networkmodel 60 comprises two portions of neural model 40, 58 as described inrelation with FIG. 4.

According to an embodiment of the present disclosure, the portion ofneural network model 60 further comprises a network array 62 having afourth number of neurons 64 and a fifth number of interneurons 68distributed among the neurons of the network array, wherein the fifthnumber is smaller than the fourth number. According to an embodiment ofthe present disclosure, the axon of each neuron 64 of the network arrayforms an excitatory STDP synapse 70 with the dendrite of a neighboringinterneuron 68 of the network array 62. According to an embodiment ofthe present disclosure, the axon of each interneuron 68 of the networkarray 62 forms an inhibitory STDP synapse 72 with the dendrite ofneighboring neurons 64 and interneurons 68 of the network array 62.According to an embodiment of the present disclosure, the axon of eachneuron 44 of the intermediate array 42 of the first neural networkportion 40 forms an excitatory STDP synapse 74 with the dendrite of aplurality of neurons 64 of the network array 62. According to anembodiment of the present disclosure, the axon of each neuron 44 of thesecond array 42 of the second neural network portion 58 forms anexcitatory STDP synapse 76 with the dendrite of a plurality of neurons64 of the network array.

According to an embodiment of the present disclosure, the network array62 comprises rows and columns of neurons 64 the axon of each neuron 44of the second array 42 of the first neural network portion 40 forms anexcitatory STDP synapse 74 with the dendrite of a plurality of neurons64 of a row of the network array 62. The axon of each neuron 44 of thesecond array 42 of the second neural network portion 58 then forms anexcitatory STDP synapse 76 with the dendrite of a plurality of neurons64 of a column of the network array 62.

According to an embodiment of the present disclosure, the axon of eachneuron 44 of the second array 42 of the first neural network portion 40forms an excitatory STDP synapse 74 with the dendrite of a plurality ofneurons 64 of a Gaussian neighborhood of neurons 64 of the network array62; and the axon of each neuron 44 of the second array 42 of the secondneural network portion 58 forms an excitatory STDP synapse 76 with thedendrite of a plurality of neurons 64 of a Gaussian neighborhood ofneurons 64 of the network array 62.

According to an embodiment of the present disclosure, the axon of eachneuron 44 of the second array 42 of the first neural network portion 40forms an excitatory STDP synapse 74 with the dendrite of a plurality ofrandom neurons 64 of the network array; and the axon of each neuron 44of the second array 42 of the second neural network portion 58 forms anexcitatory STDP synapse 76 with the dendrite of a plurality of randomneurons 64 of the network array 42.

FIG. 6 illustrates a portion of a neural network or neural network model80 according to an embodiment of the present disclosure, comprising theportion of neural network 60 described in relation with FIG. 5. Forclarity, portions 40 and 58 are not illustrated. According to anembodiment of the present disclosure, neural network 80 comprises athird neural network portion 82 as described in relation with FIG. 4,comprising an input array (not shown) arranged for receiving inputsignals, and an intermediate array 42 having neurons 44 and interneurons48. Neural network portion 82 is a training portion of neural network80. According to an embodiment of the present disclosure, neural network80 also comprises an output array 84 having a same number of neurons 86as the intermediate array 42 of portion 82. According to an embodimentof the present disclosure, output array 84 comprises interneurons 88distributed among the neurons 86. Interneurons 88 can be in the samenumber as in intermediate array 42. According to an embodiment of thepresent disclosure, the axon of each neuron 86 of the output array formsan excitatory STDP synapse 90 with the dendrite of a neighboringinterneuron 88; and the axon of each interneuron 88 of the output arrayforms an inhibitory STDP synapse 92 with the dendrite of neighboringneurons 86 and interneurons 88 of the output array. According to anembodiment of the present disclosure, the dendrite of each neuron 86 ofthe output array forms an excitatory STDP synapse 94 with the axon of aplurality of neurons 64 of the network array 62; and the dendrite ofeach neuron 86 of the output array 84 forms an excitatory non-STDPsynapse 96 with the axon of a corresponding neuron 44 of theintermediary array 42 of the neural network portion 82.

According to an embodiment of the present disclosure, the input signalsto the neural network portions 40 and 58 relate to variable parametersthat are to be correlated to input signals to training portion 82 duringa training period.

According to an embodiment of the present disclosure, after a trainingperiod, input signals are no more sent to training portion 82, and thesignals at the axon of neurons 86 of the output array provide the outputof the neural network 80 to input signals provided to the neural networkportions 40 and 58.

FIG. 7 schematizes a neural network 80 according to an embodiment of thepresent disclosure. The neural network 80 comprises a training portion82, comprising an input array/layer 12 connected to an intermediatearray/layer 42, connected as detailed above to an output array/layer 36.Neural network 80 further comprises two input portions 40 and 58 havingeach an input array/layer 12 connected to an intermediate layer 42; theintermediate layers being connected to a network layer 62. itselfconnected to output layer 84. According to an embodiment of the presentdisclosure, input portions 40 and 58 can be of identical or differentsizes. For example, an input portion having a large number of inputneurons can be used to observe a parameter with increased precision, andreciprocally.

According to an embodiment of the present disclosure, neural network 80can comprise more than one output layer 84 and more than one trainingportion such as training portion 82. Where neural network 80 comprisesan additional output layer and one or more additional training portions,having sizes identical to, or different from, output layer 84 andtraining portion 82, the additional output layers and training portionscan be connected to network layer 62 consistently with output layer 84and training portion 82. The additional training portions will thenreceive in input additional parameters to be correlated with theparameters input to portions 40 and 58 during the training period, andthe additional output layers will output said additional parameter inresponse to said parameters input to portions 40 and 58 after thetraining period.

According to an embodiment of the present disclosure, neural network 80can comprise only one input portion 40 or more input portions than thetwo input portions 40 and 58. The neural network can then comprise morethan one network layer 62, as well as intermediate network layers 62, ifappropriate. Any number of input layers may be used depending on theapplication and the desired configuration. For example, the number oflayers may reach 100 layers or more.

FIG. 8 illustrates the application of a neural network model 100according to an embodiment of the present disclosure to a planar 2DLrobotic arm 102. According to an embodiment of the present disclosure,the 2DL robotic arm 102 comprises a first arm 104 capable of making anangle θ1 with respect to a support 106 at a planar joint 108 arranged ata first end of arm 104. According to an embodiment of the presentdisclosure, the 2DL robotic arm 102 comprises a second arm 110 capableof making an angle θ2 in the same plane as θ1 with respect to the firstarm 104 at a planar joint 112 arranged at a second end of arm 104.

According to an embodiment of the present disclosure, neural networkmodel 100 comprises a first input layer L₁ ¹ coupled in a sparsefeedforward configuration via STDP synapses to a first intermediatelayer L₂ ¹, corresponding to arrays 12 and 42 of the first neuralnetwork portion 40 of FIG. 7. According to an embodiment of the presentdisclosure, neural network model 100 comprises a second input layer L₁ ²and a second intermediate layer L₂ ², corresponding to arrays 12 and 42of the second neural network portion 58 of FIG. 7.

According to an embodiment of the present disclosure, neural networkmodel 100 comprises a network layer L₃ corresponding to array 62 of FIG.7 and connected to first and second intermediate layer L₂ ¹, L₂ ².

According to an embodiment of the present disclosure, neural networkmodel 100 comprises a first training layer L₁ ^(x) and a firstintermediate layer L₂ ^(x), corresponding to arrays 12 and 42 of thetraining neural network portion 82 of FIG. 7. According to an embodimentof the present disclosure, neural network model 100 comprises a secondtraining layer L₁ ^(y) and a second intermediate layer L₂ ^(y),corresponding to arrays 12 and 42 of an additional (not shown in FIG. 7)training portion consistent with training portion 82 of FIG. 7.

According to an embodiment of the present disclosure, neural networkmodel 100 comprises a first output layer L₄ ^(x) corresponding to layer84 of FIG. 7. According to an embodiment of the present disclosure,neural network model 100 comprises a second output layer L₄ ^(y)corresponding to an additional (not shown in FIG. 7) output layerconsistent with output layer 84 of FIG. 7.

Table (a) below illustrates the number of neurons that were usedaccording to an embodiment of the present disclosure for the variouslayers/arrays of neural network model 100.

(a) Neuron Layer type Neurons L₁ ^(θ1) E 1000 L₁ ^(θ2) E 1000 L₁ ^(x) E1000 L₁ ^(y) E 1000 L₂ ^(θ1) E, I 250 L₂ ^(θ2) E, I 250 L₂ ^(x) E, I 250L₂ ^(y) E, I 250 L₃ E, I 2000 L₄ ^(x) E, I 250 L₄ ^(y) E, I 250

Further, table (b) below illustrates the type and number of synapsesexisting between the neurons of the various layers of neural networkmodel 100. According to embodiments of the present disclosure, anelectrical synapse may refer to as a mathematical model of a synapse foruse in applications including hardware, software, or a combination ofboth.

(b) Synapse Layer type Synapses L₁ ^(θ1) → L₂ ^(θ1) STDP 40 000   L₁^(θ2) → L₂ ^(θ2) STDP 40 000   L₁ ^(x) → L₂ ^(x) STDP 40 000   L₁ ^(y) →L₂ ^(y) STDP 40 000   L₂ ^(θ1) → L₂ ^(θ1) STDP 1000 L₂ ^(θ2) → L₂ ^(θ2)STDP 1000 L₂ ^(x) → L₂ ^(x) STDP 1000 L₂ ^(y) → L₂ ^(y) STDP 1000 L₂^(θ1) → L₃ STDP 230 000   L₂ ^(θ2) → L₃ STDP 230 000   L₃ → L₃ STDP 2000L₃ → L₄ ^(x) STDP 320 000   L₃ → L₄ ^(y) STDP 320 000   L₄ ^(x) → L₄^(x) STDP 1000 L₄ ^(y) → L₄ ^(y) STDP 1000 L₂ ^(x) → L₄ ^(x) Fixed,  200one-to-one L₂ ^(y) → L₄ ^(y) Fixed,  200 one-to-one

According to an embodiment of the present disclosure, input layer L₁^(θ1) and input layer L₁ ^(θ2) received input signals corresponding tothe values of angles θ1 and θ2, having spiking rates for examplecomprised between 1 Hz and 100 Hz. For example, the spiking rate of aneuron m corresponding to layer L₁ ^(θ1) was high when the angle ofjoint 108 was close to an angular position θ_(1m) associated to neuronm. According to an embodiment of the present disclosure, the spikingrates of the neighboring neurons (m−1 and m+1; etc. . . . ) responded ina Gaussian fashion with lower spiking rates farther away from neuronthat spikes maximally. It is noted that according to an embodiment ofthe present disclosure, the neurons may respond to a small range ofvalues for the variable of interest (e.g., θ₁ for L₁ ^(θ1)). The signalscorresponding to θ1 and θ2 were for example generated by proprioception,i.e from the internal state of the robotic arm.

According to an embodiment of the present disclosure, training layer L₁^(x) and input layer L₁ ^(y) received input signals corresponding to theposition of the distal end of arm 110 in the plan of motion of the arm,in a coordinate system having x and y axes. The signals corresponding tox and y were for example generated using the processing of an imagecapture of the robotic arm, with:

x=l ₁ cos(θ₁)+l ₂ cos(θ₁+θ₂)

y=l ₁ sin(θ₁)+l ₂ sin(θ₁+θ₂)

Where l₁ and l₂ are the lengths of the two arm 104, 110 of the robot. Inone embodiment, the joint angles (θ₁, θ₂) ranged from 0° to 360° whilethe x and y ranged from −1 to 1.

According to an embodiment of the present disclosure, the firing rateover time of the input and training signals can be represented by acosine or similar curve. The firing rate r may be expressed as follows:

$r = {R_{0} + {R_{1}\left( {^{\frac{- {({s - a})}^{2}}{2\sigma^{2}}} + ^{\frac{- {({s + N - a})}^{2}}{2\sigma^{2}}} + ^{\frac{- {({s - N - a})}^{2}}{2\sigma^{2}}}} \right)}}$

where R₀ is a minimum firing rate, R₁ is a maximum firing rate, σrepresents a standard deviation of neuron location that is used in theGaussian function to weight the firing rate depending upon the neuronlocation, and N is a total number of neurons in an input layer.

In one embodiment, the firing rate can be comprised between 1 Hz and 100Hz; preferably between 10 Hz and 80 Hz and σ may be 5.

According to an embodiment of the present disclosure, to compensate forvariable synaptic path lengths between the input layers from joint anglespace to L₄ and between input layers from position space to L₄ (theposition space having shorter path lengths than joint angle spacepathways to layer L₄), a delay d in the feedback pathways (i.e., L₂ ^(x)to L₄ ^(x)) may be used. In biological systems, this feedback may besimilar to a delay in the proprioceptive feedback either from a visualsystem or through additional processing in the sensory cortex.

According to an embodiment of the present disclosure, a leaky integrateand fire neuron model can be used in which a neuron receives multipleexcitatory input current signals (i₁, i₂, i₃ . . . ) and produces asingle output spike signal. The output information can be encoded intothe timing of these spikes (t₁, t₂ . . . ). The potential, V, of theleaky integrate and fire model can be determined using the membraneequation as:

$\begin{matrix}{{\tau_{m}\frac{V}{t}} = {{- \left( {V_{rest} - V} \right)} + {\sum{{w_{ex}(t)}\left( {E_{ex} - V} \right)}} - {\sum{{w_{in}(t)}\left( {E_{m} - V} \right)}}}} & (1)\end{matrix}$

with E_(ex)=0 mV and E_(in)=0 mV. When the membrane potential reaches athreshold voltage V_(thr), the neuron fires an action potential, and themembrane potential is reset to V_(rest).

According to an embodiment of the present disclosure, an integrate andfire neural cell provides several different variables to control itsmembrane voltage including synaptic conductance w (both inhibitory andexcitatory), membrane time constant τ_(m), the various constants forpotentials (e.g., E_(ex)) and threshold for firing.

Synaptic inputs to the neuron may be configured as conductance changeswith instantaneous rise times and exponential decays so that a singlepre-synaptic spike at time t generates a synaptic conductance forexcitatory and inhibitory synapses as follows:

$\begin{matrix}{{w_{ex}(t)} = {w\; ^{\frac{- t}{\tau_{AMPA}}}}} & (2) \\{{w_{in}(t)} = {w\; ^{\frac{- t}{\tau_{GABA}}}}} & (3)\end{matrix}$

where τ_(AMPA) and τ_(GABA) are the time constants forα-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPA)for excitatory neurons and gamma-aminobutyric acidGABA receptors forinhibitory synapses.

In this configuration, the neuron model may be self-normalizing in whichthe multiplicative effect of synaptic input occurs on its own membranevoltage, referred to as voltage shunting. This neuron model may enablesthe self-regulation of its own excitation and is biologicallyconsistent. The value of the excitatory synaptic conductance w_(ex)(t)(in equation 1) is determined by STDP. We will now outline the STDPlearning rule.

In one example, a synapse may be represented by a junction between twointerconnected neurons. The synapse may include two terminals. Oneterminal may be associated with the axon of the neuron providinginformation (this neuron is referred as the pre-synaptic neuron). Theother terminal may be associated with the dendrite of the neuronreceiving the information (this is referred as the post-synapticneuron).

For a synapse with a fixed synaptic conductance, w, only the input andthe output terminals may be required. In one example, the conductance ofthe synapse may be internally adjusted according to a learning rulereferred to as the spike-timing dependent plasticity or STDP.

The system may be configured with a STDP function that modulates thesynaptic conductance w based on the timing difference(ti_(pre)−tj_(post)) between the action potentials of pre-synapticneuron i and post-synaptic neuron j. There are two possibilities for themodulation of synaptic conductance. If the timing difference(ti_(pre)−tj_(post)) is positive, then synapse undergoes depression. Ifthe timing difference (ti_(pre)−tj_(post)) is negative, then the synapsemay undergo potentiation. If the timing difference is too large oneither direction, there is no change in the synaptic conductance. In oneembodiment, the timing difference may be 80 ms.

The STDP function may include four parameters (A+, A−, τ⁺ and τ⁻) thatcontrol the shape of the function. The A+ and A− correspond to themaximum change in synaptic conductance for potentiation and depressionrespectively. The time constants τ⁺ and τ⁻ control the rate of decay forpotentiation and depression portions of the curve as shown in FIG. 5(a).

In one method, more than one pre- or post-synaptic spike within the timewindows for potentiation or depression may occur. Accounting for thesemultiple spikes may be performed using an additive STDP model where thedynamics of potentiation P and depression D at a synapse are governedby:

$\begin{matrix}{{\tau^{-}\frac{D}{t}} = {- D}} & (4) \\{{\tau^{+}\frac{P}{t}} = {- P}} & (5)\end{matrix}$

Whenever a post-synaptic neuron fires a spike, D is decremented by anamount A⁻ relative to the value governed by equation (6). Similarly,every time a synapse receives a spike from a pre-synaptic neuron, P isincremented by an amount A⁺ relative to value governed by equation (7).These changes may be summarized as:

D=D+A ⁻  (6)

P=P+A ⁺  (7)

These changes to P and D may affect the change in synaptic conductance.If the post-synaptic neuron fires a spike, then the value of P at thattime, P*, is used to increment Δw for the duration of that spike.Similarly, if the pre-synaptic neuron fires a spike that is seen by thesynapse, then the value of D at that time, D*, is used to decrement Δwfor the duration of that spike. Thus, the net change Δw is given by:

Δw=P*−D*  (8)

The final effective change to the synaptic conductance w due to STDP maybe expressed as:

w=w+Δw  (9)

In one embodiment as shown in FIG. 8, a spiking model may be configuredto learn multiple transformations from a fixed set of input spiketrains. As shown in FIG. 6, several prediction layer neurons 624 may becoupled or connected to their own training outputs 614. A predictionlayer may refer to an output set of neurons 622 that predict theposition of a robotic arm. In one embodiment, the model in FIG. 6 mayfunction similar to the model described in FIG. 1.

In another embodiment, the system 600 described below may simultaneouslylearn multiple outputs or transformations of the input spike trains. Inone example, the same inputs angles (θ1, θ2) may be used by the spikingmodel to generate multiple outputs using the equations 10 and 11 in thefollowing.

The inventors have shown that a model as illustrated in FIG. 8 may beconfigured to learn several types of functions including anticipation,association and prediction and inverse transformations. In oneembodiment, the system may be configured to use multiple possiblepathways for input-to-output transformations. As discussed hereafter,the model is also fault tolerant.

FIG. 9 illustrates the synaptic conductances between various layers ofneural network model 100 during learning showing the emergence oftopological organization of conductances in the neural network model ofFIG. 8.

FIG. 10A illustrates the output of layer L₄ ^(y) at a given time t inresponse to inputs on layers L₁ ^(θ1) and L₁ ^(θ2), after the trainingperiod of the neural network 100 has been completed. The diameter of thecircles on the y, θ1 and θ2 axes increases with the firing rate of theneurons forming each axis. Only the neurons that fire are illustrated.

According to an embodiment of the disclosure, decoding the output of aneural network 80 as illustrated in FIG. 7 comprises:

a/ providing the first arrays 12 of the first and second neural networkportions 40, 58 with first and second input signals having a rate thatincreases when a measured parameter gets closer to a predetermined valueassigned to the neurons of said first arrays;

b/ assigning to each neuron of the output array of neurons 84 anincremental position value comprised between 1 and N, N being the numberof neurons of the output array 84;

c/ at any given time, measuring the firing rate of each neuron of theoutput array 84; and

d/ estimating the output of the neural network, at said any given time,as corresponding to the neuron of the output array 84 having a positionvalue equal to a division of the sum of the position value of eachneuron of the output array, weighted by its firing rate at said anygiven time, by the sum of the firing rates of each neuron of the outputarray at said any given time.

In other terms,

${y_{P}\left( {i,j,t} \right)} = \frac{\sum\limits_{k = 1}^{N}{{f_{ijk}(t)} \cdot {y\left( {i,j,k,t} \right)}}}{\sum\limits_{k = 1}^{N}{f_{ijk}(t)}}$

with y_(p)(i, j, t) the evaluated output position at a given time t, forgiven values i, j of θ1 and θ2; f_(ijk)(t) being the firing rate for aneuron k, at time t, for given values i, j of θ1 and θ2; and y(i, j, k,t) being the position value of a neuron k at time t, for given values i,j of θ1 and θ2.

FIG. 10B illustrates the output of layer L₄ ^(y) at a given time t inresponse to inputs on layers L₁ ^(θ1) and L₁ ^(θ2), after the trainingperiod of the neural network 100 has been completed, where the output onthe y axis wraps around the end of the output array. According to anembodiment of the present disclosure, the method of measuring the outputof array 84 comprises, if the neurons of the middle of the output array84 have null firing rates, assigning to the neurons of lower positionvalue a position value increased by the value N, N being the number ofneurons of the output array 84. According to an embodiment of thepresent disclosure, the method described in relation with FIGS. 10A and10B can also be used to decode the output of for example layer 84 ofFIG. 14, described hereafter.

FIGS. 11A-C illustrate the incremental convergence of the neural networkmodel of FIG. 8 as a function of learning. In particular, FIG. 11Aillustrates the x and y output of the neural network model of FIG. 8after a training period of 300 seconds; FIG. 11B illustrates the x and youtput of the neural network model of FIG. 8 after a training period of600 seconds; and FIG. 11C illustrates the x and y output of the neuralnetwork model of FIG. 8 after a training period of 1500 seconds. Thereal values of x, y corresponding to the inputs used for FIGS. 11A-Cfollow the pretzel-shaped trajectory shown in darker.

FIGS. 12A-B illustrate the incremental convergence of the neural networkmodel of FIG. 8 when a Gaussian sparse connectivity is used between theneurons 44 of the intermediate arrays 42 and the network array 62.

FIGS. 12C-D illustrate the incremental convergence of the neural networkmodel of FIG. 8 when a random sparse connectivity is used between theneurons 44 of the intermediate arrays 42 and the network array 62.

FIGS. 13A-D illustrate the performances of the neural network model ofFIG. 8 for varying degrees of damaged neurons. FIG. 13A(a) shows theneural activity, or cortical coding, of the synapses between L₁ ^(θ1)and L₂ ^(θ1) for a network having 5% of neurons damaged. FIG. 13A(b)shows the neural activity, or cortical coding, of the synapses within L₂^(θ1) for a network having 5% of neurons damaged. FIG. 13A(c) shows theoutput x, y of a network having 5% of neurons damaged, compared to thereal values of x, y (darker circle) corresponding to the inputs used forproducing the output.

FIG. 13B(a)(b)(c) shows the same data as FIG. 13A(a)(b)(c) for networkhaving 8% of neurons damaged.

FIG. 13C(a)(b)(c) shows the same data as FIG. 13A(a)(b)(c) for networkhaving 12% of neurons damaged.

FIG. 13D(a)(b)(c) shows the same data as FIG. 13A(a)(b)(c) for networkhaving 16% of neurons damaged.

As illustrated by FIGS. 13A-D, a neural network according to anembodiment of the present disclosure is robust to neuron damage, andproduces satisfactory output even with significant neuron damage.

FIG. 14 illustrates a portion of a neural network or neural networkmodel 118 according to an embodiment of the present disclosure,comprising an input array 12 of neurons 14 coupled to an intermediatearray 42 of neurons 44 and interneurons 48. According to an embodimentof the present disclosure, the input array/layer 12 comprises first andsecond sub-arrays 120 and 122 of neurons 14. The neurons 14 of the firstsub-array 120 are provided for receiving input signals related to afirst measured parameter. The neurons 14 of the second sub-array 122 areprovided for receiving input signals related to a second measuredparameter. According to an embodiment of the present disclosure, theintermediate array 42 comprises rows and columns of neurons 44; theinterneurons 48 being distributed among the neurons, wherein the axon ofeach neuron 14 of the first sub-array of neurons 120 forms an excitatorySTDP synapse with the dendrite of a plurality of neurons 44 of a row ofthe intermediate array 42; and wherein the axon of each neuron 14 of thesecond sub-array of neurons 122 forms an excitatory STDP synapse withthe dendrite of a plurality of neurons 44 of a column of theintermediate array 42.

According to an embodiment of the present disclosure, the neurons 44 ofthe intermediate array 42 can be arranged according to another schemenot comprising rows and columns; or the neurons of the first and secondsub arrays 102, 122 can be connected to the neurons of intermediatearray 42 according to a scheme, for example a sparse and randomconnection scheme, not following rows and columns in intermediate array42. According to an embodiment of the present disclosure, one dendriteof a neuron 44 of the intermediate array 42 can form STDP synapses withthe axon of 100 to 200 neurons 14 of the input array. According to anembodiment of the present disclosure, sub arrays 120, 122 can eachcomprise 1000 neurons and intermediate array can comprise 2000 neurons.

According to an embodiment of the present disclosure, input array 12 cancomprise a number N of sub-arrays of neurons such as 120, 122,respectively provided for receiving input signals related to a number Nof associated measured parameters. According to an embodiment of thepresent disclosure, each neuron 14 of each sub-array is provided forreceiving an input signal indicating that the measured parameterassociated to the sub-array gets closer to a predetermined valueassigned to said neuron. For example, the rate of the signal sent to aneuron can increase when the measured parameter gets closer to apredetermined value assigned to said neuron, and reciprocally. Thenumber of neurons of the sub-arrays can be identical or different.

According to an embodiment of the present disclosure, the neurons areintegrate and fire neurons, or operate under a model of integrate andfire neurons and the neural network or neural network model is a spikingneural network or spiking neural network model.

According to an embodiment of the present disclosure, neural network 118comprises an output array 84 having neurons 86 and interneurons 88distributed among the neurons 86. According to an embodiment of thepresent disclosure, output array 84 can comprise one interneuron 88 forfour neurons 86. According to an embodiment of the present disclosure,the axon of each neuron 86 of the output array forms an excitatory STDPsynapse 90 with the dendrite of the neighboring interneurons 88; and theaxon of each interneuron 88 of the output array forms an inhibitory STDPsynapse 92 with the dendrite of the neighboring neurons 86 andinterneurons 88 of the output array.

According to an embodiment of the present disclosure, the dendrite ofeach neuron 86 of the output array 84 forms an excitatory STDP synapsewith the axon of each neuron 44 of the intermediate array 42.

According to an embodiment of the present disclosure, neural network 118comprises a training array 124 comprising as many neurons 126 as theoutput array 84.

According to an embodiment of the present disclosure, the dendrite ofeach neuron 126 is provided for receiving an input signal indicatingthat a measured parameter gets closer to a predetermined value assignedto said neuron. According to an embodiment of the present disclosure,the axon of each neuron 126 of the training array 124 forms anexcitatory non-STDP synapse with the dendrite of a corresponding neuronof the output array 84.

According to an embodiment of the present disclosure, the input signalsto the first and second sub-arrays 120, 122 relate to variableparameters that are to be correlated by the neural network to theparameter that relate to the input signals to the training array 124.According to an embodiment of the present disclosure, the parametersignals are sent to first and second sub-arrays 120, 122 as well as totraining array 124 during a training period. The signals sent to firstand second sub-arrays 120, 122 can for example correspond to two anglesmeasured for a two-level of freedom robot arm such as shown in FIG. 8,for random positions of the arm, whereas the signal sent to trainingarray 124 can for example correspond to an x or y coordinate of theposition of an end of said robot arm, as measured for each of saidrandom positions.

After the training period, input signals are no more sent to trainingarray 124 and the signals at the axon of neurons 86 of the output arrayprovide the output of the neural network 118 to input signals providedto input arrays 120, 122.

FIG. 15 illustrates the portion of a neural network or neural networkmodel 118 of FIG. 14, comprising additional output layers 128, 130connected to intermediate layer 42 in the same way as output layer 84.According to an embodiment of the present disclosure, output layers 84,128 and 130 can comprise the same number of neurons, or differentnumbers of neurons. According to an embodiment of the presentdisclosure, additional output layers 128, 130 are connected to traininglayers 132, 134 in the same way output layer 84 is connected to traininglayer 124. According to an embodiment of the present disclosure, neuralnetwork 118 can comprise any number of additional output layer, eachoutput layer being connected to a training layer as detailed above. Thetraining periods for each output layer can have the same length and besimultaneous, or they can have different lengths and/or happen atdifferent times.

FIG. 16 illustrates a portion of a neural network or neural networkmodel 150 comprising the neural network portion 118 of FIG. 15.According to an embodiment of the present disclosure, network 150comprises additional neural network portions 152, 154, 156 similar toneural network portion 118, wherein the training arrays 134, 124, 132 ofneural network portion 118 also from an input layer or input sub-arrayof neural network portions 152, 154, 156. According to an embodiment ofthe present disclosure, a training array 158 of neural network portion152 forms an input sub-array of neural network portion 154. Neuralnetwork portions 118, 152, 154, 156 can be of the same size or ofdifferent sizes. Network 150 can comprise any number of neural networkportions such as neural network portion 118.

In embodiments of the present disclosures, the neural network may beimplemented using a shared processing device, individual processingdevices, or a plurality of processing devices. Such a processing devicemay be a microprocessor, micro-controller, digital signal processor,microcomputer, central processing unit, field programmable gate array,programmable logic device, state machine, logic circuitry, analogcircuitry, digital circuitry, and/or any device that manipulates signals(analog and/or digital) based on operational instructions.

The present disclosure or any part(s) or function(s) thereof, may beimplemented using hardware, software, or a combination thereof, and maybe implemented in one or more computer systems or other processingsystems. A computer system for performing the operations of the presentdisclosure and capable of carrying out the functionality describedherein can include one or more processors connected to a communicationsinfrastructure (e.g., a communications bus, a cross-over bar, or anetwork). Various software embodiments are described in terms of such anexemplary computer system. After reading this description, it willbecome apparent to a person skilled in the relevant art(s) how toimplement the disclosure using other computer systems and/orarchitectures.

The foregoing description of the preferred embodiments of the presentdisclosure has been presented for purposes of illustration anddescription. It is not intended to be exhaustive or to limit thedisclosure to the precise form or to exemplary embodiments disclosed.Obviously, many modifications and variations will be apparent topractitioners skilled in this art. Similarly, any process stepsdescribed might be interchangeable with other steps in order to achievethe same result. The embodiment was chosen and described in order tobest explain the principles of the disclosure and its best modepractical application, thereby to enable others skilled in the art tounderstand the disclosure for various embodiments and with variousmodifications as are suited to the particular use or implementationcontemplated. It is intended that the scope of the disclosure be definedby the claims appended hereto and their equivalents. Reference to anelement in the singular is not intended to mean “one and only one”unless explicitly so stated, but rather means “one or more.” Moreover,no element, component, nor method step in the present disclosure isintended to be dedicated to the public regardless of whether theelement, component, or method step is explicitly recited in thefollowing claims. No claim element herein is to be construed under theprovisions of 35 U.S.C. Sec. 112, sixth paragraph, unless the element isexpressly recited using the phrase “means for . . . .”

It should be understood that the figures illustrated in the attachments,which highlight the functionality and advantages of the presentdisclosure, are presented for example purposes only. The architecture ofthe present disclosure is sufficiently flexible and configurable, suchthat it may be utilized (and navigated) in ways other than that shown inthe accompanying figures.

Furthermore, the purpose of the foregoing Abstract is to enable the U.S.Patent and Trademark Office and the public generally, and especially thescientists, engineers and practitioners in the art who are not familiarwith patent or legal terms or phraseology, to determine quickly from acursory inspection the nature and essence of the technical disclosure ofthe application. The Abstract is not intended to be limiting as to thescope of the present disclosure in any way. It is also to be understoodthat the steps and processes recited in the claims need not be performedin the order presented.

The various features of the present disclosure can be implemented indifferent systems without departing from the present disclosure. Itshould be noted that the foregoing embodiments are merely examples andare not to be construed as limiting the present disclosure. Thedescription of the embodiments is intended to be illustrative, and notto limit the scope of the claims. As such, the present teachings can bereadily applied to other types of apparatuses and many alternatives,modifications, and variations will be apparent to those skilled in theart.

1. A neural network, wherein a portion of the neural network comprises:a first array having a first number of neurons, wherein the dendrite ofeach neuron of the first array is provided for receiving an input signalindicating that a measured parameter gets closer to a predeterminedvalue assigned to said neuron; a second array having a second number ofneurons, the dendrite of each neuron of the second array forming anexcitatory STDP synapse with the axon of a plurality of neurons of thefirst array; and the dendrite of each neuron of the second array formingan excitatory STDP synapse with the axon of neighboring neurons of thesecond array.
 2. The neural network of claim 1, wherein the secondnumber is smaller than the first number.
 3. The neural network of claim1, wherein the second array further comprises a third number ofinterneurons distributed among the neurons of the second array, whereinthe third number is smaller than the second number, wherein: the axon ofeach neuron of the second array forms an excitatory STDP synapse withthe dendrite of the neighboring interneurons of the second array; andthe axon of each interneuron of the second array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the second array.
 4. The neural network of claim 1,wherein the dendrite of each neuron of the first array is provided forreceiving an input signal having a rate that increases when a measuredparameter gets closer to a predetermined value assigned to said neuron.5. A neural network comprising a first and a second neural networkportions according to claim 1; and a third array having a fourth numberof neurons and a fifth number of interneurons distributed among theneurons of the third array, wherein the fifth number is smaller than thefourth number, wherein: the axon of each neuron of the third array formsan excitatory STDP synapse with the dendrite of the neighboringinterneurons of the third array; and the axon of each interneuron of thethird array forms an inhibitory STDP synapse with the dendrite of theneighboring neurons and interneurons of the third array; wherein theaxon of each neuron of the second array of the first neural networkportion forms an excitatory STDP synapse with the dendrite of aplurality of neurons of the third array; and wherein the axon of eachneuron of the second array of the second neural network portion forms anexcitatory STDP synapse with the dendrite of a plurality of neurons ofthe third array.
 6. The neural network of claim 5, wherein the thirdarray comprises rows and columns of neurons, wherein the axon of eachneuron of the second array of the first neural network portion forms anexcitatory STDP synapse with the dendrite of a plurality of neurons of arow of the third array; and wherein the axon of each neuron of thesecond array of the second neural network portion forms an excitatorySTDP synapse with the dendrite of a plurality of neurons of a column ofthe third array.
 7. The neural network of claim 5, comprising a thirdneural network portion according to claim 1, as well as a fourth arrayhaving a second number of neurons and a third number of interneuronsdistributed among the neurons of the fourth array, wherein: the axon ofeach neuron of the fourth array forms an excitatory STDP synapse withthe dendrite of the neighboring interneurons of the fourth array; andthe axon of each interneuron of the fourth array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the fourth array; wherein the dendrite of each neuron ofthe fourth array forms an excitatory STDP synapse with the axon of aplurality of neurons of the third array; and wherein the dendrite ofeach neuron of the fourth array forms an excitatory non-STDP synapsewith the axon of a corresponding neuron of the second array of the thirdneural network.
 8. The neural network of claim 7, wherein the inputsignals to the first and second neural network portions relate tovariable parameters that are to be correlated to the input signals tothe third neural network.
 9. The neural network of claim 1, wherein saidfirst array of neurons comprises first and second sub-arrays of neuronsprovided for receiving input signals related to first and secondmeasured parameters, respectively.
 10. The neural network of claim 9,wherein the second array comprises rows and columns of neurons; whereinthe axon of each neuron of the first sub-array of neurons forms anexcitatory STDP synapse with the dendrite of a plurality of neurons of arow of the second array; and wherein the axon of each neuron of thesecond sub-array of neurons forms an excitatory STDP synapse with thedendrite of a plurality of neurons of a column of the second array. 11.The neural network of claim 10, wherein the second array furthercomprises a third number of interneurons distributed among the neuronsof the second array, wherein the third number is smaller than the secondnumber, wherein: the axon of each neuron of the second array forms anexcitatory STDP synapse with the dendrite of the neighboringinterneurons of the second array; and the axon of each interneuron ofthe second array forms an inhibitory STDP synapse with the dendrite ofthe neighboring neurons and interneurons of the second array.
 12. Theneural network of claim 11; further comprising: a third array having afourth number of neurons and a fifth number of interneurons distributedamong the neurons of the third array, wherein the fifth number issmaller than the fourth number, wherein: the axon of each neuron of thethird array forms an excitatory STDP synapse with the dendrite of theneighboring interneurons of the third array; and the axon of eachinterneuron of the third array forms an inhibitory STDP synapse with thedendrite of the neighboring neurons and interneurons of the third array;wherein the dendrite of each neuron of the third array forms anexcitatory STDP synapse with the axon of each neuron of the secondarray.
 13. The neural network of claim 12, comprising a fourth arraycomprising as many neurons as the third array of neurons, wherein thedendrite of each neuron of the fourth array is provided for receiving aninput signal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron; wherein the axon of eachneuron of the fourth array forms an excitatory non-STDP synapse with thedendrite of a corresponding neuron of the third array.
 14. The neuralnetwork of claim 13, wherein the input signals to the first and secondsub-arrays of neurons relate to variable parameters that are to becorrelated to the input signals to the fourth array.
 15. The neuralnetwork of claim 13, wherein the fourth array of neurons is a sub-arrayof neurons of a further neural network according to claim
 9. 16. Amethod of programming a neural network, comprising: providing a firstneural network portion comprising a first array having a first number ofneurons and a second array having a second number of neurons, thedendrite of each neuron of the second array forming an excitatory STDPsynapse with the axon of a plurality of neurons of the first array; thedendrite of each neuron of the second array forming an excitatory STDPsynapse with the axon of neighboring neurons of the second array; andproviding to the dendrite of each neuron of the first array an inputsignal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron.
 17. The method of claim 16,further comprising providing the second array with a third number ofinterneurons distributed among the neurons of the second array, whereinthe third number is smaller than the second number, wherein: the axon ofeach neuron of the second array forms an excitatory STDP synapse withthe dendrite of the neighboring interneurons of the second array; andthe axon of each interneuron of the second array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the second array.
 18. The method of claim 17, comprisingproviding the dendrite of each neuron of the first array with an inputsignal having a rate that increases when a measured parameter getscloser to a predetermined value assigned to said neuron.
 19. The methodof claim 17, comprising: providing a second neural network portionhaving the same structure as the first neural network portion; andproviding a third array having a fourth number of neurons and a fifthnumber of interneurons distributed among the neurons of the third array,wherein the fifth number is smaller than the fourth number, wherein: theaxon of each neuron of the third array forms an excitatory STDP synapsewith the dendrite of the neighboring interneurons of the third array;and the axon of each interneuron of the third array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the third array; wherein the axon of each neuron of thesecond array of the first neural network portion forms an excitatorySTDP synapse with the dendrite of a plurality of neurons of the thirdarray; and wherein the axon of each neuron of the second array of thesecond neural network portion forms an excitatory STDP synapse with thedendrite of a plurality of neurons of the third array; and providing tothe dendrite of each neuron of the first array of the second neuralnetwork portion an input signal indicating that a measured parametergets closer to a predetermined value assigned to said neuron.
 20. Themethod of claim 19, comprising: providing a third neural network portionhaving the same structure as the first neural network portion; providinga fourth array having a second number of neurons and a third number ofinterneurons distributed among the neurons of the fourth array, wherein:the axon of each neuron of the fourth array forms an excitatory STDPsynapse with the dendrite of the neighboring interneurons of the fourtharray; and the axon of each interneuron of the fourth array forms aninhibitory STDP synapse with the dendrite of the neighboring neurons andinterneurons of the fourth array; wherein the dendrite of each neuron ofthe fourth array forms an excitatory STDP synapse with the axon of aplurality of neurons of the third array; and wherein the dendrite ofeach neuron of the fourth array forms an excitatory non-STDP synapsewith the axon of a corresponding neuron of the second array of the thirdneural network; and providing to the dendrite of each neuron of thefirst array of the third neural network portion an input signalindicating that a measured parameter gets closer to a predeterminedvalue assigned to said neuron; wherein the input signals to the firstand second neural network portions relate to variable parameters thatare to be correlated to the input signals to the third neural networkportion.
 21. The method of claim 16, wherein: said providing to thedendrite of each neuron of the first array an input signal indicatingthat a measured parameter gets closer to a predetermined value assignedto said neuron comprises: providing to the dendrite of each neuron of afirst subset of neurons of the first array an input signal indicatingthat a first measured parameter gets closer to a predetermined valueassigned to said neuron; providing to the dendrite of each neuron of asecond subset of neurons of the first array an input signal indicatingthat a second measured parameter gets closer to a predetermined valueassigned to said neuron.
 22. The method of claim 21, wherein: saidproviding a second array having a second number of neurons comprisesproviding a second array having rows and columns of neurons, wherein theaxon of each neuron of the first subset of neurons of the first arrayforms an excitatory STDP synapse with the dendrite of a plurality ofneurons of a row of the second array; and wherein the axon of eachneuron of the second subset of neurons of the first array forms anexcitatory STDP synapse with the dendrite of a plurality of neurons of acolumn of the second array.
 23. The method of claim 21, furthercomprising providing the second array with a third number ofinterneurons distributed among the neurons of the second array, whereinthe third number is smaller than the second number, wherein: the axon ofeach neuron of the second array forms an excitatory STDP synapse withthe dendrite of the neighboring interneurons of the second array; andthe axon of each interneuron of the second array forms an inhibitorySTDP synapse with the dendrite of the neighboring neurons andinterneurons of the second array.
 24. The method of claim 23,comprising: providing a third array having a fourth number of neuronsand a fifth number of interneurons distributed among the neurons of thethird array, wherein the fifth number is smaller than the fourth number,wherein the axon of each neuron of the third array forms an excitatorySTDP synapse with the dendrite of the neighboring interneurons of thethird array; and the axon of each interneuron of the third array formsan inhibitory STDP synapse with the dendrite of the neighboring neuronsand interneurons of the third array; wherein the dendrite of each neuronof the third array forms an excitatory STDP synapse with the axon ofeach neuron of the second array; and providing a fourth array comprisingas many neurons as the third array of neurons, wherein the dendrite ofeach neuron of the fourth array is provided for receiving an inputsignal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron; and wherein the axon ofeach neuron of the fourth array forms an excitatory non-STDP synapsewith the dendrite of a corresponding neuron of the third array; andproviding to the dendrite of each neuron of the fourth array an inputsignal indicating that a measured parameter gets closer to apredetermined value assigned to said neuron; wherein the input signalsto the first and second subset of neurons relate to variable parametersthat are to be correlated to the input signals to the fourth array. 25.A method of decoding an output of a neural network according to claim 8;the method comprising: providing the first arrays of the first andsecond neural network portions with first and second input signalshaving a rate that increases when a measured parameter gets closer to apredetermined value assigned to the neurons of said first arrays;assigning to each neuron of the fourth array of neurons an incrementalposition value comprised between 1 and N, N being the number of neuronsof the fourth array; at any given time, measuring the firing rate ofeach neuron of the fourth array; and estimating the output of the neuralnetwork, at said any given time, as corresponding to the neuron of thefourth array having a position value equal to a division of the sum ofthe position value of each neuron of the fourth array, weighted by itsfiring rate at said any given time, by the sum of the firing rates ofeach neuron of the fourth array at said any given time.
 26. The methodof claim 25 comprising, if the neurons of the middle of the fourth arrayhave null firing rates, assigning to the neurons of lower position valuea position value increased by the value N.
 27. A method of decoding anoutput of a neural network according to claim 14; the method comprising:providing the first and second sub-arrays of neurons with first andsecond input signals having a rate that increases when a measuredparameter gets closer to a predetermined value assigned to the neuronsof said first and second sub-arrays of neurons; assigning to each neuronof the third array of neurons an incremental position value comprisedbetween 1 and N, N being the number of neurons of the third array; atany given time, measuring the firing rate of each neuron of the thirdarray; and estimating the output of the neural network, at said anygiven time, as corresponding to the neuron of the third array having aposition value equal to a division of the sum of the position value ofeach neuron of the third array, weighted by its firing rate at said anygiven time, by the sum of the firing rates of each neuron of the thirdarray at said any given time.
 28. The method of claim 27 comprising, ifthe neurons of the middle of the third array have null firing rates,assigning to the neurons of lower position value a position valueincreased by the value N.